c# pdf free : C# read pdf from url Library SDK class asp.net .net wpf ajax TRENCH_REAL_ANALYSIS0-part225

INTRODUCTION
TOREALANALYSIS
WilliamF.Trench
AndrewG.CowlesDistinguishedProfessorEmeritus
DepartmentofMathematics
TrinityUniversity
SanAntonio,Texas,USA
wtrench@trinity.edu
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FUNCTIONSDEFINEDBYIMPROPERINTEGRALS
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LibraryofCongressCataloging-in-PublicationData
Trench,WilliamF.
Introductiontorealanalysis / / WilliamF.Trench
p.cm.
ISBN0-13-045786-8
1.MathematicalAnalysis. I.
Title.
QA300.T6672003
515-dc21
2002032369
FreeHyperlinkedEdition2.04December2013
ThisbookwaspublishedpreviouslybyPearsonEducation.
Thisfreeeditionismadeavailableinthehopethatitwillbeusefulasatextbookorrefer-
ence. Reproductionispermittedforanyvalidnoncommercialeducational,mathematical,
orscientificpurpose. However, , chargesforprofitbeyondreasonableprintingcostsare
prohibited.
Acompleteinstructor’ssolutionmanualisavailablebyemailtowtrench@trinity.edu,sub-
jecttoverificationoftherequestor’sfacultystatus. Althoughthisbookissubjecttoa
CreativeCommonslicense,thesolutionsmanualisnot. Theauthorreservesallrightsto
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Contents
Preface
vi
Chapter1 TheRealNumbers
1
1.1 TheReal l NumberSystem
1
1.2 MathematicalInduction
10
1.3 TheReal l Line
19
Chapter2 DifferentialCalculusofFunctionsofOne e Variable30
2.1 FunctionsandLimits
30
2.2 Continuity
53
2.3 DifferentiableFunctionsofOne e Variable
73
2.4 L’Hospital’sRule
88
2.5 Taylor’sTheorem
98
Chapter3 IntegralCalculusofFunctions s ofOneVariable
113
3.1 DefinitionoftheIntegral
113
3.2 ExistenceoftheIntegral
128
3.3 PropertiesoftheIntegral
135
3.4 ImproperIntegrals
151
3.5 AMore e AdvancedLookattheExistence
ofthe ProperRiemannIntegral
171
Chapter4 Infinite e SequencesandSeries
178
4.1 SequencesofRealNumbers
179
4.2 EarlierTopicsRevisitedWithSequences
195
4.3 Infinite e SeriesofConstants
200
iv
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Contents
v
4.4 SequencesandSeriesofFunctions
234
4.5 PowerSeries
257
Chapter5 Real-ValuedFunctionsofSeveralVariables
281
5.1 Structureof
RRR
n
281
5.2 Continuous s Real-ValuedFunctionof
n
Variables
302
5.3 PartialDerivativesandthe e Differential
316
5.4 TheChainRuleandTaylor’sTheorem
339
Chapter6 Vector-ValuedFunctionsofSeveralVariables
361
6.1 LinearTransformationsandMatrices
361
6.2 ContinuityandDifferentiabilityofTransformations
378
6.3 TheInverseFunctionTheorem
394
6.4. The e ImplicitFunctionTheorem
417
Chapter7 IntegralsofFunctionsofSeveralVariables
435
7.1 DefinitionandExistenceoftheMultipleIntegral
435
7.2 IteratedIntegralsandMultipleIntegrals
462
7.3 ChangeofVariablesinMultiple e Integrals
484
Chapter8 MetricSpaces
518
8.1 IntroductiontoMetricSpaces
518
8.2 CompactSetsinaMetricSpace
535
8.3 Continuous s FunctionsonMetricSpaces
543
AnswerstoSelectedExercises
549
Index
563
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Preface
Thisisatextforatwo-termcourseinintroductoryrealanalysisforjuniororseniormath-
ematicsmajorsandsciencestudentswithaseriousinterestinmathematics. Prospective
educatorsormathematicallygiftedhighschoolstudentscanalsobenefitfromthemathe-
maticalmaturitythatcanbegainedfromanintroductoryrealanalysiscourse.
Thebookisdesignedtofillthegapsleftinthedevelopmentofcalculusasitisusually
presentedinanelementarycourse,andtoprovidethebackgroundrequiredforinsightinto
moreadvancedcoursesinpureandappliedmathematics. Thestandardelementarycalcu-
lussequenceistheonlyspecificprerequisiteforChapters1–5,whichdealwithreal-valued
functions.(However,otheranalysisorientedcourses,suchaselementarydifferentialequa-
tion, alsoprovideusefulpreparatoryexperience.) ) Chapters6and7requireaworking
knowledgeofdeterminants,matricesandlineartransformations,typicallyavailablefroma
firstcourseinlinearalgebra.Chapter8isaccessibleaftercompletionofChapters1–5.
Withouttakingapositionfororagainstthecurrentreformsinmathematicsteaching,I
thinkitisfairtosaythatthetransitionfromelementarycoursessuchascalculus,linear
algebra,anddifferentialequationstoarigorousrealanalysiscourseisabiggerstepto-
daythanitwasjustafewyearsago. Tomakethissteptoday’sstudentsneedmorehelp
thantheirpredecessorsdid,andmustbecoachedandencouragedmore. Therefore,while
strivingthroughouttomaintainahighlevelofrigor,Ihavetriedtowriteasclearlyandin-
formallyaspossible.InthisconnectionIfinditusefultoaddressthestudentinthesecond
person. Ihaveincluded295completelyworkedoutexamplestoillustrateandclarifyall
majortheoremsanddefinitions.
Ihaveemphasizedcarefulstatementsofdefinitionsandtheoremsandhavetriedtobe
completeanddetailedinproofs,exceptforomissionslefttoexercises. Igiveathorough
treatmentofreal-valuedfunctionsbeforeconsideringvector-valuedfunctions. Inmaking
thetransitionfromonetoseveralvariablesandfromreal-valuedtovector-valuedfunctions,
Ihavelefttothestudentsomeproofsthatareessentiallyrepetitionsofearliertheorems. I
believethatworkingthroughthedetailsofstraightforwardgeneralizationsofmoreelemen-
taryresultsisgoodpracticeforthestudent.
Greatcare has goneintothepreparationofthe761numberedexercises, , manywith
multipleparts. Theyrangefromroutinetoverydifficult. Hintsareprovidedforthemore
difficultpartsoftheexercises.
vi
Preface
vii
Organization
Chapter1isconcernedwiththerealnumbersystem. Section1.1beginswithabriefdis-
cussionoftheaxiomsforacompleteorderedfield,butnoattemptismadetodevelopthe
realsfromthem;rather,itisassumedthatthestudentisfamiliarwiththeconsequencesof
theseaxioms,exceptforone:completeness. Sincethedifferencebetweenarigorousand
nonrigoroustreatmentofcalculuscanbedescribedlargelyintermsoftheattitudetaken
towardcompleteness,Ihavedevotedconsiderableefforttodevelopingitsconsequences.
Section1.2isaboutinduction. Althoughthismayseem m outofplaceinarealanalysis
course,Ihavefoundthatthetypicalbeginningrealanalysisstudentsimplycannotdoan
inductionproofwithoutreviewingthemethod.Section1.3isdevotedtoelementarysetthe-
oryandthetopologyoftherealline,endingwiththeHeine-BorelandBolzano-Weierstrass
theorems.
Chapter2coversthedifferentialcalculusoffunctionsofonevariable:limits,continu-
ity,differentiablility,L’Hospital’srule,andTaylor’stheorem.Theemphasisisonrigorous
presentationofprinciples;noattemptismadetodevelopthepropertiesofspecificele-
mentaryfunctions. Eventhoughthismaynotbedonerigorouslyinmostcontemporary
calculuscourses,Ibelievethatthestudent’stimeisbetterspentonprinciplesratherthan
onreestablishingfamiliarformulasandrelationships.
Chapter3istodevotedtotheRiemannintegraloffunctionsofonevariable. InSec-
tion3.1theintegralisdefinedinthestandardwayintermsofRiemannsums. Upperand
lowerintegralsarealsodefinedthereandusedinSection3.2tostudytheexistenceofthe
integral.Section3.3isdevotedtopropertiesoftheintegral.Improperintegralsarestudied
inSection3.4. Ibelievethatmytreatmentofimproperintegralsismoredetailedthanin
mostcomparabletextbooks.AmoreadvancedlookattheexistenceoftheproperRiemann
integralisgiveninSection3.5,whichconcludeswithLebesgue’sexistencecriterion.This
sectioncanbeomittedwithoutcompromisingthestudent’spreparednessforsubsequent
sections.
Chapter4treatssequencesandseries. Sequences s ofconstantarediscussed d inSec-
tion4.1. Ihavechosentomaketheconceptsoflimitinferiorandlimitsuperiorparts
ofthisdevelopment,mainlybecausethispermitsgreaterflexibilityandgenerality,with
littleextraeffort,inthestudyofinfiniteseries. Section4.2providesabriefintroduction
tothewayinwhichcontinuityanddifferentiabilitycanbestudiedbymeansofsequences.
Sections4.3–4.5treatinfiniteseriesofconstant,sequencesandinfiniteseriesoffunctions,
andpowerseries,againingreaterdetailthaninmostcomparabletextbooks. Theinstruc-
torwhochoosesnottocoverthesesectionscompletelycanomitthelessstandardtopics
withoutlossinsubsequentsections.
Chapter5isdevotedtoreal-valuedfunctionsofseveralvariables. Itbeginswithadis-
cussionofthetoplogyofR
n
inSection5.1.Continuityanddifferentiabilityarediscussed
inSections5.2and5.3.ThechainruleandTaylor’stheoremarediscussedinSection5.4.
viii
Preface
Chapter6coversthedifferentialcalculusofvector-valuedfunctionsofseveralvariables.
Section6.1reviewsmatrices,determinants,andlineartransformations,whichareintegral
partsofthedifferentialcalculusaspresentedhere. InSection6.2thedifferentialofa
vector-valuedfunctionisdefinedasalineartransformation,andthechainruleisdiscussed
intermsofcompositionofsuchfunctions.Theinversefunctiontheoremisthesubjectof
Section6.3,wherethenotionofbranchesofaninverseisintroduced. InSection6.4. the
implicitfunctiontheoremismotivatedbyfirstconsideringlineartransformationsandthen
statedandprovedingeneral.
Chapter7coverstheintegralcalculusofreal-valuedfunctionsofseveralvariables.Mul-
tipleintegralsaredefinedinSection7.1,firstoverrectangularparallelepipedsandthen
overmoregeneralsets.Thediscussiondealswiththemultipleintegralofafunctionwhose
discontinuitiesformasetofJordancontentzero. Section7.2dealswiththeevaluationby
iteratedintegrals.Section7.3beginswiththedefinitionofJordanmeasurability,followed
byaderivationoftheruleforchangeofcontentunderalineartransformation,anintuitive
formulationoftheruleforchangeofvariablesinmultipleintegrals,andfinallyacareful
statementandproofoftherule.Theproofiscomplicated,butthisisunavoidable.
Chapter8dealswithmetricspaces. Theconceptandpropertiesofametricspaceare
introducedinSection8.1. Section8.2discussescompactnessinametricspace,andSec-
tion8.3discussescontinuousfunctionsonmetricspaces.
Corrections–mathematicalandtypographical–arewelcomeandwillbeincorporatedwhen
received.
WilliamF.Trench
wtrench@trinity.edu
Home:659HopkintonRoad
Hopkinton,NH03229
CHAPTER1
TheRealNumbers
INTHISCHAPTERwebeginthestudyoftherealnumbersystem.Theconceptsdiscussed
herewillbeusedthroughoutthebook.
SECTION1.1dealswiththeaxiomsthatdefinetherealnumbers, definitionsbasedon
them,andsomebasicpropertiesthatfollowfromthem.
SECTION1.2emphasizestheprincipleofmathematicalinduction.
SECTION1.3 introducesbasicideas s ofset theoryinthecontextofsets ofrealnum-
bers. Inthissectionweprovetwofundamentaltheorems:theHeine–BorelandBolzano–
Weierstrasstheorems.
1.1THEREALNUMBERSYSTEM
Havingtakencalculus,youknowalotabouttherealnumbersystem;however,youprob-
ablydonotknowthatallitspropertiesfollowfromafewbasicones. Althoughwewill
notcarryoutthedevelopmentoftherealnumbersystemfromthesebasicproperties,itis
usefultostatethemasastartingpointforthestudyofrealanalysisandalsotofocuson
oneproperty,completeness,thatisprobablynewtoyou.
FieldProperties
Therealnumbersystem(whichwewilloftencallsimplythereals)isfirstofallaset
fa;b;c;:::gonwhichtheoperationsofadditionandmultiplicationaredefinedsothat
everypairofrealnumbershasauniquesumandproduct,bothrealnumbers, withthe
followingproperties.
(A)
aCbDbCaandabDba(commutativelaws).
(B)
.aCb/CcDaC.bCc/and.ab/cDa.bc/(associativelaws).
(C)
a.bCc/DabCac(distributivelaw).
(D)
Therearedistinctrealnumbers0and1suchthataC0Daanda1Dafor alla.
(E)
ForeachathereisarealnumberasuchthataC.a/D0,andifa¤0, thereis
arealnumber1=asuchthata.1=a/D1.
1
2 Chapter1
TheRealNumbers
Themanipulativepropertiesoftherealnumbers,suchastherelations
.aCb/
2
Da
2
C2abCb
2
;
.3aC2b/.4cC2d/D12acC6adC8bcC4bd;
.a/D.1/a; a.b/D.a/bDab;
and
a
b
C
c
d
D
adCbc
bd
.b;d ¤0/;
allfollowfrom
(A)
(E)
.Weassumethatyouarefamiliarwiththeseproperties.
Asetonwhichtwooperationsaredefinedsoastohaveproperties
(A)
(E)
iscalleda
field.Therealnumbersystemisbynomeanstheonlyfield.Therationalnumbers(which
aretherealnumbersthatcanbewrittenasrDp=q,wherepandqareintegersandq¤0)
alsoformafieldunderadditionandmultiplication.Thesimplestpossiblefieldconsistsof
twoelements,whichwedenoteby0and1,withadditiondefinedby
0C0D1C1D0; 1C0D0C1D1;
(1.1.1)
andmultiplicationdefinedby
00D01D10D0; 11D1
(1.1.2)
(Exercise1.1.2).
TheOrderRelation
Therealnumbersystemisorderedbytherelation<,whichhasthefollowingproperties.
(F)
Foreachpairofrealnumbersaandb,exactlyoneofthefollowingistrue:
aDb; a<b; or b<a:
(G)
Ifa<bandb<c,thena<c.(Therelation<istransitive.)
(H)
Ifa<b,thenaCc<bCcforanyc,andif0<c,thenac<bc.
Afieldwithanorderrelationsatisfying
(F)
(H)
isanorderedfield. Thus,thereal
numbersformanorderedfield. Therationalnumbersalsoformanorderedfield,butitis
impossibletodefineanorderonthefieldwithtwoelementsdefinedby(1.1.1)and(1.1.2)
soastomakeitintoanorderedfield(Exercise1.1.2).
Weassumethatyouarefamiliarwithotherstandardnotationconnectedwiththeorder
relation:thus,a>bmeansthatb<a;abmeansthateitheraDbora>b;ab
meansthateithera D bora <b;theabsolutevalueof a, denotedbyjaj,equalsaif
a0oraifa0.(Sometimeswecalljajthemagnitudeofa.)
Youprobablyknowthefollowingtheoremfromcalculus,butweincludetheprooffor
yourconvenience.
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